function 207

For this reason, mathematicians say that there are only

seven possible frieze patterns.

A wallpaper pattern is the two-dimensional ana-

logue of a frieze pattern. Mathematicians have proved

that there are exactly 17 different wallpaper patterns.

frustum If a geometric solid is cut by two parallel

planes, then the portion of the solid between the two

planes is called a frustum. A frustum is also produced

by cutting the solid by one plane parallel to the

BASE

of

the solid. For example, a conical frustum, is the shape

produced by slicing the top off a cone (with the cut

made parallel to the base). A pyramidal frustum is

made by slicing off the top of a pyramid, again with

the cut made parallel to the base.

The altitude hof a frustum is the distance between

the two parallel planes. The volume of a conical or

pyramidal frustum is given by the formula:

where A1and A2are the areas of the upper and lower

faces.

See also

CONE

;

SOLID OF REVOLUTION

;

VOLUME

.

F-test See

STATISTICS

:

INFERENTIAL

.

function (mapping) Any procedure or a rule that

assigns to each member of one set Xone, and only one,

element of another set Yis called a function. For exam-

ple, the relation “is the mother of” is a function from

the set Xof all the people of the world to the set Yof

all women: each person is “assigned” one, and only

one, biological mother. The rule that squares numbers

takes members of the set X= {1,2,3,4} to correspond-

ing elements of the set Y= {1,4,9,16}.

If fis a symbol used to denote a function from a set

Xto a set Y, then we write f: X →Yand call Xthe

domain of fand Ythe codomain of f. We also say that f

“maps” elements of Xto elements of Y. If a specific ele-

ment xof the set Xis mapped to the element yin Y, then

we write y= f(x).(Mathematicians sometimes call xan

“input” and the value ythe corresponding “output.”)

For instance, if fis the squaring function described

above, then we have f(2) = 4 and f(3) = 9. If “M”

denotes the “is the mother of” function, and Lawrence’s

mother is Trenyce, then we can write M(Lawrence) =

Trenyce. Sometimes mathematicians will use arrows

with tabs to emphasize the notion of a mapping and

write, for example, M: Lawrence |

→Trenyce.

Notice that not all elements of Yneed be the result

of a mapping. (Not all women are mothers, for exam-

ple.) Also, it is possible for two or more elements of X

to be mapped to the same element of Y. (Two people

could have the same mother, for instance.) The set of

all possible outputs f(x) for a given function fis called

the range (or the

IMAGE

) of the function. For example,

the range of the “is the mother of” function M, is the

set of all women who have given birth.

In mathematics one typically studies functions

between sets of numbers, usually described by formu-

lae. For example, the squaring function is described by

the equation f(x) = x2, or just y= x2. The function y=

3x+ 2, for example, describes the rule “assign to any

number xthe number ytwo more than three times x.”

In this context, xis called the independent variable and

ythe dependent variable. (Its value is dependent on the

value of x.) A

GRAPH OF A FUNCTION

is a pictorial rep-

resentation of a function that operates on numbers.

Equations, such as y= x2and y= 3x+ 2, in which

the dependent variable is expressed in terms of the

independent variable, are said to be in “explicit form.”

In contrast, expressions for which both the indepen-

dent and dependent variables appear on one side of the

equality sign are said to be in “implicit form.” For

instance, the equations 2x– y= 5 and x2+ y2= 9 are in

implicit form. Implicit equations might, or might not,

define yas a function of x. For example, the first equa-

tion presented above can be rewritten y= 2x– 5, yield-

ing the explicit form of a function. The second relation,

however, does not represent a function: there is no

unique value of yassociated to each value of x. (For

example, with x= 0, ycould be either 3 or –3.)

Any set of ordered pairs (x,y) satisfying some

given constraint is called a relation. For example, the

set of all ordered pairs (x,y) satisfying the equation x2

+ y2= 9 is a relation. Graphically, this relation repre-

sents a circle of radius three. A relation is a function if

it is never the case that two distinct ordered pairs have

the same x-coordinate value (that is, no single x-value

has two different y-values associated to it). This condi-

tion is sometimes called the “vertical line test.” For